AP Calculus BC Flashcards: Calculating Higher Order Derivatives

What this deck covers

This deck focuses on Calculating Higher Order Derivatives, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the fourth derivative of $f(x) = x^4$?

Answer: $f^{(4)}(x) = 24$. Fourth derivative of $x^4$ gives the factorial $4! = 24$.

Flashcard 2: What is the second derivative of $f(x) = \text{sin}^2(x)$?

Answer: $f''(x) = 2\text{cos}(2x)$. Use double angle identity: $\sin^2(x) = \frac{1-\cos(2x)}{2}$.

Flashcard 3: What is the second derivative of $f(x) = \text{e}^{2x}$?

Answer: $f''(x) = 4\text{e}^{2x}$. Chain rule with exponential: derivative multiplies by the inner function coefficient squared.

Flashcard 4: What is the second derivative of $f(x) = x \text{ln}(x)$?

Answer: $f''(x) = \frac{1}{x}$. Use product rule: $f'(x) = \ln(x) + 1$, then $f''(x) = \frac{1}{x}$.

Flashcard 5: State the formula for the second derivative of $f(x) = \text{ln}(x)$.

Answer: $f''(x) = -\frac{1}{x^2}$. Standard formula for logarithmic second derivative.

Flashcard 6: What is the second derivative of $f(x) = e^x$?

Answer: $f''(x) = e^x$. Derivative of $e^x$ is always $e^x$.

Flashcard 7: Find the fourth derivative of $f(x) = x^4 - 3x^2 + 2$.

Answer: $f^{(4)}(x) = 24$. Fourth derivative of $x^4$ term is $24$, other terms become zero.

Flashcard 8: Compute the third derivative of $f(x) = \text{e}^{-2x}$.

Answer: $f'''(x) = -8\text{e}^{-2x}$. Chain rule with $e^{-2x}$: coefficient becomes $(-2)^3 = -8$.

Flashcard 9: What is the second derivative of $f(x) = x^2 \text{e}^x$?

Answer: $f''(x) = (x^2 + 4x + 2)\text{e}^x$. Product rule applied twice to $x^2 e^x$.

Flashcard 10: State the second derivative of $f(x) = \text{cos}(x)$.

Answer: $f''(x) = -\text{cos}(x)$. Second derivative of $\cos(x)$ follows trig cycle pattern.

Flashcard 11: What is the third derivative of $f(x) = \text{sin}(2x)$?

Answer: $f'''(x) = -8\text{sin}(2x)$. Chain rule with $\sin(2x)$: coefficient becomes $(-2)^3 = -8$.

Flashcard 12: Find the third derivative of $f(x) = \frac{1}{3}x^3$.

Answer: $f'''(x) = 2$. Third derivative eliminates the coefficient $\frac{1}{3}$ leaving constant $2$.

Flashcard 13: Find the third derivative of $f(x) = 5x^4 + 3x^3 - x$.

Answer: $f'''(x) = 120x + 18$. Apply power rule to each term and differentiate three times.

Flashcard 14: What is the second derivative of $f(x) = \text{e}^{3x}$?

Answer: $f''(x) = 9\text{e}^{3x}$. Chain rule with $e^{3x}$: coefficient becomes $3^2 = 9$.

Flashcard 15: Find the second derivative of $f(x) = \text{e}^{-x}$.

Answer: $f''(x) = \text{e}^{-x}$. Derivative of $e^{-x}$ multiplies by $(-1)^2 = 1$.

Flashcard 16: What is the third derivative of $f(x) = \text{sin}(2x)$?

Answer: $f'''(x) = -8\text{sin}(2x)$. Chain rule with $\sin(2x)$: coefficient becomes $(-2)^3 = -8$.

Flashcard 17: What is the second derivative of $f(x) = \frac{1}{x}$?

Answer: $f''(x) = \frac{2}{x^3}$. Rewrite as $x^{-1}$, then apply power rule twice.

Flashcard 18: What is the second derivative of $f(x) = \text{e}^{3x}$?

Answer: $f''(x) = 9\text{e}^{3x}$. Chain rule with $e^{3x}$: coefficient becomes $3^2 = 9$.

Flashcard 19: What is the second derivative of $f(x) = \text{e}^{2x}$?

Answer: $f''(x) = 4\text{e}^{2x}$. Chain rule with exponential: derivative multiplies by the inner function coefficient squared.

Flashcard 20: Compute the fourth derivative of $f(x) = \text{cos}(3x)$.

Answer: $f^{(4)}(x) = 81\text{cos}(3x)$. Fourth derivative of $\cos(3x)$ involves $3^4 = 81$ and returns to cosine.

Flashcard 21: Compute the third derivative of $f(x) = \frac{1}{x^2}$.

Answer: $f'''(x) = \frac{6}{x^5}$. Rewrite as $x^{-2}$ and apply power rule three times.

Flashcard 22: Compute the third derivative of $f(x) = x^2 \text{sin}(x)$.

Answer: $f'''(x) = -2(3\text{cos}(x) + x\text{sin}(x))$. Use product rule repeatedly on $x^2 \sin(x)$.

Flashcard 23: Find the third derivative of $f(x) = 5x^4 + 3x^3 - x$.

Answer: $f'''(x) = 120x + 18$. Apply power rule to each term and differentiate three times.

Flashcard 24: What is the second derivative of $f(x) = x^3$?

Answer: $f''(x) = 6x$. Apply power rule twice: $f'(x) = 3x^2$, then $f''(x) = 6x$.

Flashcard 25: What is the second derivative of $f(x) = x \text{ln}(x)$?

Answer: $f''(x) = \frac{1}{x}$. Use product rule: $f'(x) = \ln(x) + 1$, then $f''(x) = \frac{1}{x}$.

Flashcard 26: State the second derivative of $f(x) = \text{tan}(x)$.

Answer: $f''(x) = 2\text{sec}^2(x)\text{tan}(x)$. First derivative is $\sec^2(x)$, apply chain rule again.

Flashcard 27: Compute the third derivative of $f(x) = \text{sin}(x)$.

Answer: $f'''(x) = -\text{cos}(x)$. Trig derivatives cycle: $\sin \to \cos \to -\sin \to -\cos$.

Flashcard 28: Find the second derivative of $f(x) = e^{-x}$.

Answer: $f''(x) = e^{-x}$. Derivative of $e^{-x}$ multiplies by $(-1)^2 = 1$.

Flashcard 29: State the second derivative of $f(x) = \text{tan}(x)$.

Answer: $f''(x) = 2\text{sec}^2(x)\text{tan}(x)$. First derivative is $\sec^2(x)$, apply chain rule again.

Flashcard 30: What is the second derivative of $f(x) = \text{ln}(x^2)$?

Answer: $f''(x) = -\frac{2}{x^2}$. Use chain rule: $\ln(x^2) = 2\ln(x)$, so derivative doubles.

Flashcard 31: What is the fourth derivative of $f(x) = x^4$?

Answer: $f^{(4)}(x) = 24$. Fourth derivative of $x^4$ gives the factorial $4! = 24$.

Flashcard 32: Find the fourth derivative of $f(x) = x^4 - 3x^2 + 2$.

Answer: $f^{(4)}(x) = 24$. Fourth derivative of $x^4$ term is $24$, other terms become zero.

Flashcard 33: Compute the third derivative of $f(x) = \text{sin}(x)$.

Answer: $f'''(x) = -\text{cos}(x)$. Trig derivatives cycle: $\sin \to \cos \to -\sin \to -\cos$.

Flashcard 34: Compute the fourth derivative of $f(x) = \text{cos}(3x)$.

Answer: $f^{(4)}(x) = 81\text{cos}(3x)$. Fourth derivative of $\cos(3x)$ involves $3^4 = 81$ and returns to cosine.

Flashcard 35: What is the fourth derivative of $f(x) = x^4 - x^2 + 1$?

Answer: $f^{(4)}(x) = 24$. Fourth derivative eliminates lower-order terms, leaving only $x^4$ coefficient.

Flashcard 36: What is the second derivative of $f(x) = \frac{1}{x}$?

Answer: $f''(x) = \frac{2}{x^3}$. Rewrite as $x^{-1}$, then apply power rule twice.

Flashcard 37: What is the second derivative of $f(x) = \text{sin}^2(x)$?

Answer: $f''(x) = 2\text{cos}(2x)$. Use double angle identity: $\sin^2(x) = \frac{1-\cos(2x)}{2}$.

Flashcard 38: Compute the third derivative of $f(x) = \frac{1}{x^2}$.

Answer: $f'''(x) = \frac{6}{x^5}$. Rewrite as $x^{-2}$ and apply power rule three times.

Flashcard 39: What is the second derivative of $f(x) = e^x$?

Answer: $f''(x) = e^x$. Derivative of $e^x$ is always $e^x$.

Flashcard 40: Compute the third derivative of $f(x) = x^2 \text{sin}(x)$.

Answer: $f'''(x) = -2(3\text{cos}(x) + x\text{sin}(x))$. Use product rule repeatedly on $x^2 \sin(x)$.

Flashcard 41: Find the third derivative of $f(x) = \frac{1}{3}x^3$.

Answer: $f'''(x) = 2$. Third derivative eliminates the coefficient $\frac{1}{3}$ leaving constant $2$.

Flashcard 42: What is the fourth derivative of $f(x) = x^4 - x^2 + 1$?

Answer: $f^{(4)}(x) = 24$. Fourth derivative eliminates lower-order terms, leaving only $x^4$ coefficient.

Flashcard 43: State the formula for the second derivative of $f(x) = \text{ln}(x)$.

Answer: $f''(x) = -\frac{1}{x^2}$. Standard formula for logarithmic second derivative.

Flashcard 44: State the second derivative of $f(x) = \text{cos}(x)$.

Answer: $f''(x) = -\text{cos}(x)$. Second derivative of $\cos(x)$ follows trig cycle pattern.

Flashcard 45: What is the second derivative of $f(x) = x^2 \text{e}^x$?

Answer: $f''(x) = (x^2 + 4x + 2)\text{e}^x$. Product rule applied twice to $x^2 e^x$.

Flashcard 46: Compute the third derivative of $f(x) = \text{e}^{-2x}$.

Answer: $f'''(x) = -8\text{e}^{-2x}$. Chain rule with $e^{-2x}$: coefficient becomes $(-2)^3 = -8$.

Flashcard 47: What is the second derivative of $f(x) = \text{arcsin}(x)$?

Answer: $f''(x) = \frac{x}{(1-x^2)^{3/2}}$. Inverse trig derivatives involve radical expressions in denominators.

Flashcard 48: What is the second derivative of $f(x) = \text{arcsin}(x)$?

Answer: $f''(x) = \frac{x}{(1-x^2)^{3/2}}$. Inverse trig derivatives involve radical expressions in denominators.

Flashcard 49: What is the second derivative of $f(x) = \text{ln}(x)$?

Answer: $f''(x) = -\frac{1}{x^2}$. First derivative is $\frac{1}{x}$, second is $-\frac{1}{x^2}$.